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Computation of fundamental groups of Galois covers recently led to the construction and analysis of Coxeter covers of the symmetric groups [L. H. Rowen, M. Teicher and U. Vishne, Coxeter covers of the symmetric groups, J. Group Theory8 (2005) 139-169]. In this paper we consider analog covers of Artin's braid groups, and completely describe the induced geometric extensions of the braid group. [ABSTRACT FROM AUTHOR]

This is the final paper in a series of four, concerning the surface 핋 × 핋 embedded in ℂℙ8, where 핋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface. [ABSTRACT FROM AUTHOR]

This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is $\mathbb{Z}_{c}^{n-2}$ where c = gcd(a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface F1(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1(2, 2) with respect to a generic projection is isomorphic to $\mathbb{Z}_2^{10}$. [ABSTRACT FROM AUTHOR]

Let X be the surface 𝕋 × 𝕋, where 𝕋 is the complex torus. This article is the third in a series studying the fundamental group of the Galois cover of X with respect to a generic projection onto ℂℙ 2 . Van Kampen Theorem gives a presentation of the fundamental group of the complement of the branch curve, with 54 generators and more than 2000 relations. Here we introduce a certain natural quotient (obtained by identifying pairs of generators), prove it is a quotient of a Coxeter group related to the degeneration of X , and show that this quotient is virtually nilpotent. Communicated by C. Pedrini. [ABSTRACT FROM AUTHOR]

For a maximal separable subfield K of a central simple algebra A , we provide a semiring isomorphism between K – K -sub-bimodules of A and H – H -sub-bisets of G = Gal ( L / F ) , where F = Cent ( A ) , L is the Galois closure of K / F , and H = Gal ( L / K ) . This leads to a combinatorial interpretation of the growth of dim K ⁡ ( ( K a K ) i ) , for fixed a ∈ A , especially in terms of Kummer subspaces. [ABSTRACT FROM AUTHOR]